In a triangle, the sum of lengths of two sides is x and the product of the lengths of the …

Question

In a triangle, the sum of lengths of two sides is x and the product of the lengths of the same two sides is y. If x2 -- c2 = y, where c is the length of the third side of the triangle, then the circumradius of the triangle is :

Accepted Answer

Given a + b = x   and    ab = y

If  x2 - c2 = y $$ \Rightarrow $$ (a + b)2 - c2 = ab

$$ \Rightarrow $$  a2 + b2 - c2 = - ab

$$ \Rightarrow $$   $${{{a^2} + {b^2} - {c^2}} \over {2ab}} =  - {1 \over 2}$$

$$ \Rightarrow \cos C =  - {1 \over 2}$$

$$ \Rightarrow \angle C = {{2\pi } \over 3}$$

$$R = {c \over {2\sin C}} = {c \over {\sqrt 3 }}$$

Solutions · 1